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In functional analysis, the open mapping theorem, also known as the Banach–Schauder theorem (named after Stefan Banach and Juliusz Schauder), is a fundamental result which states that if a continuous linear operator between Banach spaces is surjective then it is an open map. More precisely, : : Open Mapping Theorem. If ''X'' and ''Y'' are Banach spaces and ''A'' : ''X'' → ''Y'' is a surjective continuous linear operator, then ''A'' is an open map (i.e. if ''U'' is an open set in ''X'', then ''A''(''U'') is open in ''Y''). The proof uses the Baire category theorem, and completeness of both ''X'' and ''Y'' is essential to the theorem. The statement of the theorem is no longer true if either space is just assumed to be a normed space, but is true if ''X'' and ''Y'' are taken to be Fréchet spaces. ==Consequences== The open mapping theorem has several important consequences: * If ''A'' : ''X'' → ''Y'' is a bijective continuous linear operator between the Banach spaces ''X'' and ''Y'', then the inverse operator ''A''−1 : ''Y'' → ''X'' is continuous as well (this is called the bounded inverse theorem). * If ''A'' : ''X'' → ''Y'' is a linear operator between the Banach spaces ''X'' and ''Y'', and if for every sequence (''xn'') in ''X'' with ''xn'' → 0 and ''Axn'' → ''y'' it follows that ''y'' = 0, then ''A'' is continuous (the closed graph theorem). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Open mapping theorem (functional analysis)」の詳細全文を読む スポンサード リンク
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